Trustworthiness in Modeling Unreinforced and Reinforced T-Joints with Finite Elements
3. Present Modeling
The load-ovalization curves are presented, where the ovalization indicates the deformation of the chord cross section at the center of the T-joint, which is calculated by subtracting the height of point fixed simultaneously at the bottom of the chord and at the center of the joint,δB, from the average value of four fixed points (in the one quarter FE model of course will be only two) on the brace near the intersection with the chord,δA. These points are obtained by the transducer reading in case of experimental investigation, whereas for the FE model by using RP (Reference Point) as shown in Figure1. The ultimate load for each test is identified simply by selecting the peak of each curve, and these values are listed in Table3, whereFu,testis for the experimental investigations [4] while Fu,numfor the numerical simulations [5]. At the same time, the two phases of the joint behavior, such as the linear phase and the plastic one, can be deducted.
Table 3.Ultimate load obtained in the numerical simulation [5].
Fu,test
[kN] F[u,numkN] Fu,num/Fu,test
EX−01 305.1 310.9 1.02
EX−02 543.2 557.8 1.03
EX−03 425.6 431.4 1.01
EX−04 609.2 648.9 1.07
EX−05 780.0 798.5 1.02
EX−06 1065.3 1069.3 1.00
EX−07 415.8 446.4 1.07
EX−08 708.0 712.2 1.01
EX−09 200.1 193.0 0.96
EX−10 407.8 393.7 0.97
EX−11 305.0 309.3 1.01
EX−12 520.0 506.2 0.97
The experimental [4] and numerical [5] results are not only evaluated in terms of load-ovalization curves but also by comparison of deformed sliced “rings”, taken at the center of the T-joints after the tests and numerical analyses were completed. Then at the end of the tests, for some of the specimens, sections or “rings” were obtained for the joint elements, chord and brace.
In conclusion, all the results obtained from these two papers will be used in the fundamental step of calibration of the FE model of the present work. In the following section the results of the present FE simulations will be compared to the ones of the previous studies and comments and highlights will be provided.
With the help of preliminary simulations, the more fitting values of these have been demonstrated to beE=210 GPa andν=0.3. These values were obtained by comparing the normalized loadR, defined asR=Fu,test/fy,0·t20of the EX-01, wheret0is the chord thickness, with the one given by the experimental investigations, as reported in Table4.
Table 4.Values ofEandνof preliminary simulations [17].
Specimen Experimental Investigation
Preliminary Simulation 1 E= 203ν= 0.25
Preliminary Simulation 2 E= 210ν= 0.25
Preliminary Simulation 3 E= 210ν= 0.30 fy,0[N/mm2
285 285 285 285
t0[mm] 8.1 8.1 8.1 8.1
R=Fu,test/fy,0·t20 16.32 13.95 14.09 14.11
AutoCAD 3D was selected to draw the T-joint tubular elements. Although ABAQUS gives the possibility to directly represents different geometries, AutoCAD is a more suitable software for this kind of parametric research since it gives the possibility to easily modify and change elements on it.
Afterwards, only a quarter of the joint has been imported in the FE packages, this idealization does not make any differences in the results and reduces the total computational effort.
Starting from the first adjustments of the simulations, the three cases are discussed below:
unreinforced, collar plate reinforcement, and doubler plate reinforcement.
Three native parts have been created for the reinforced cases (chord, brace, and reinforcement plate), and only two for the unreinforced case because no reinforcement is present. Some internal parts have been introduced in the part module, with their respective edges. In other words, due to the fact that the joint is made of several parts and contact must be implemented in order to connect the different parts together, construction edges (from CAD) are kept in the FE model also in order to simplify the following step.
Once all the parts are created (in their own coordinate system), they should be assembled in the Abaqus Assembly module [51] that is used to create instances of the parts and to position the instances relative to each other in a global coordinate system. The instances made for all the three cases are one for each part. Then for the unreinforced case, two instances have been generated, one for the chord and the other for the brace; whereas for the reinforced cases a third instance that represents the reinforcement is present.
The displacement induced by the controlled actuator in the experimental investigation [4] is at an initial stroke rate of 0.3 mm/min, and then progressively increased up to 1.2 mm/min through the Instron 8800 controller. During the simulations in the present study, two types of step sequences are analyzed: the first automatic step sequence, and the second fixed step sequence, i.e., varying the load range during the simulations.
The automatic step sequence has the same step range for the whole simulation. While the fixed ones in order to reproduce the experimental investigation have two different ranges, in particular the second step starts at the starting point of the material plasticization which has been identified during the automatic step sequence.
The restraints that have been adopted for the FE model are in accordance with the experimental investigation; where the specimen is pin-supported at the chord ends and the brace end is bolted to the Instron actuator mounted on top of the brace, where the load is obtained as reaction force from the boundary condition at the brace tip. Since one-quarter of the whole specimen has been modeled, along with symmetry planes, peculiar boundary conditions have been considered as XSYMM (along the x-axis) and ZSYMM (along thez-axis) [51].
Material properties of the elements are modeled in accordance to the experimental investigation as listed in Table2, whereas the Young’s modulus and Poisson ratio are taken from the preliminary simulations already described Table4. The true stress-strain curve that has been assumed by var
der Vegte et al. [5] in his numerical simulations is represented by a multilinear relationship and subsequently converted into a true stress-true strain relationship, and no further hardening in the true stress-true strain behavior is assumed after the peak load, i.e., the curve is assumed horizontal beyond this point.
Since no more detailed information is given about the true strain-stress curve, in this study, the elastic-perfectly plastic model is assumed to fit these characteristics as shown in Figure2a. Due to the lack in the reference paper [5] for what concerns the latter relationship, in this research some hardening will be considered during the simulations, by introducing a hardening of 10% for all the materials, as depicted in Figure2b, to verify the reliability of the elastic-perfectly plastic model.
Currently, the following types of constraints have been used in the simulations, the multi-point constraints (MPC) [51], which allows the restriction imposed by the boundary conditions (BCs) along the whole circumference, and the TIE [51], which instead has been used to connect the edges of the elements to simulate the welded connections. However, since the brace is directly welded to the chord in the unreinforced cases, no separation will occur between the two elements and the two parts have been merged (i.e., combine two elements in one instance [51]) into a single one.
During the development of the contact between the chord and the reinforcement made for the reinforced T-joints, contact mechanical properties are imposed by fixing the relevant points, such as tangential and normal behavior in the contact interaction propriety. The tangential behavior was imposed as “Frictionless”, while the normal behavior as “Hard” contact. The contact properties are also imposed to “Allow separation after contact” since, in the case of the reinforced plate, separation between the plate and the chord that are linked, by the “tie” constraints, might occur. After the contact property is defined, surface-to-surface contact interaction is set up. Here the sliding is finite, and no adjustments are required since the two surfaces lie in the same plane. A further important feature is “Contact controls” option, which helps the convergence of the simulation without considering the penetration of one member into the other, as after solved using the Top/Bottom surface function [51].
The final results will be given in terms of load-ovalization curves; the ovalization was detected by the use of the transducers, so four reference points around the brace near the brace-chord intersection precisely at 20 mm distance, and one placed under the chord at the center of the T-joint have been introduced. The load detection is done by inserting an additional point at the tip of the brace, the reaction force at that point is indeed the opposite of the load applied. For the sake of further comparison, the deformed shape at the center of the joint will be considered
The FE mesh is a key point in modeling, since it has a huge influence on the results of the simulations in these kinds of elements, as will be seen when analyzing the results. The mesh module [51] allows generating meshes of parts and assemblies. For each FE model, four mesh densities are generated, and for each of them four types of elements are considered.
The four mesh densities analyzed are:
• Coarse mesh (average size 50 mm)
• Medium mesh (average size 20 mm)
• Fine mesh (almost the same number of elements as the medium ones but the size decreases from the ends to the intersection)
• Article mesh (same number of nodes on the edges shown in [5]) The four element types are:
• S4R (4 nodes with reduced integration)
• S4 (4 nodes without reduced integration)
• S8R (8 nodes with reduced integration and six degrees of freedom (DOF))
• S8R5 (8 nodes with reduced integration and five DOF)
In the “Mesh controls”, the “Structured” technique with element shape as “Quad-dominated” [51]
was used for all the simulations. These two selections are able to carry out a regular mesh made (mostly) by quadrilaterals.
The four aforementioned meshes are used in this work to better understand the fitting of the results depending on the element size. In three of the four densities cases, the medium, fine, and article one, shown in Figure3b–d, the doubling of the lines in the reinforcement area can be noticed. This is because the contact interaction algorithm needs two different meshes in order to have a reliable result of the contact [51]. In particular, the “slave surface” should have a finer mesh with respect to the
“master surface”.
(a)ȱ (b)ȱ
ȱ ȱ
(c)ȱ (d)ȱ
Figure 3.Meshes used in the simulations and their nomenclature: (a) coarse mesh, (b) medium mesh, (c) fine mesh, and (d) article mesh.
The coarse and medium mesh densities are obtained by imposing a unique size on the instance of the T-joint. The larger mesh has a size of 50, whereas the medium has a global size of 20, as depicted respectively in Figure3a,b. The total number of elements is equal to 466 in coarse mesh, while 2745 elements are generated for the medium mesh. In the remaining cases, i.e., the fine and the article meshes depicted respectively in Figure3c,d, the element sizes are varied in such a way that relatively smaller elements are used where the stress gradient is more critical. Therefore, the mesh density decreases from the vicinity of the intersection to the end of the brace or chord, and therefore smaller sizes in the vicinity of the brace-chord intersection. This strategy, to decrease the size while
getting close to the critical area, is frequently used in FE analysis, because it allows to have an accurate study of the area most influential thanks to the small sizes, and at the same time to reduce the computational effort, due to increase of the sizes far from the critical point. Fine mesh has almost the same number of elements of the medium mesh, but their sizes, as aforementioned, are decreasing while going close to the intersection. For the article mesh, as in the fine one, the sizes are smaller near the intersection, this mesh is obtained from the mesh used by var der Vegte et al. [5]. Precisely, an equal number of nodes at each edge with almost the same factor of variation (decrease in size) is considered, so a total of 341 elements is obtained. In particular, by using the proportion between the geometries of the specimens in Table1and the graphic dimensions displayed by var der Vegte et al. [5], it was possible to discretize the FE model as depicted in Figure4, in order to have the same nodes of the paper [5] at edges as previously stated.
ȱ
(a)ȱ (b)ȱ (c)ȱ
Figure 4.True article mesh density subdivision: (a) chord, (b) brace, and (c) reinforcement. [3]
From Figure4a–c, it can be seen that the reinforcement and its respective area in the chord have different repartition, in particular three of the edges of the chord are increased by two units, i.e., from six to eight spaces. This is done to help the contact interaction as aforementioned. As a result, a total of 341 elements is obtained. For the final view of the specimen in the case of the article mesh please consider Figure3d.